Good
Bad
Which competition structure is best?
What is considered “best”?
A good tournament: top k teams produced by the tournament matches closely to the true top k
Which competition structure is best?
What is considered “best”?
A good tournament: top k teams produced by the tournament matches closely to the true top k
Effectivity1: Interested in correctly ranking the best competitor as first
Efficacy2: Final rankings should match the true rankings of all the competitors
The notation for mutual information is: \[ I(X;Y) = \sum_{y \in Y}\sum_{x \in X} p(x,y) \log{\frac{p(x,y)}{p(x)p(y)}} \]
In our case, it can be simplified to:
\[ I(r, \hat{r}) = n! \sum_{\hat{r}} p(r, \hat{r}) \log{\frac{p(r,\hat{r})}{(\frac{1}{n!})^2}} \]
\[\tau_w = \frac{\sum_{i\neq j} W_{ij} \times \text{sgn}(i-j)\times \text{sgn}(R_i - R_j)}{\sum_{i,j} W_{ij} - \sum_i W_{ii}}\]
| r | theta | r_hat | Wins | Losses |
|---|---|---|---|---|
| 1 | 1.221 | 3 | 2 | 1 |
| 2 | 0.765 | 4 | 1 | 2 |
| 3 | 0.431 | 2 | 2 | 1 |
| 4 | 0.140 | 1 | 3 | 0 |
| 5 | -0.140 | 7 | 0 | 1 |
| 6 | -0.431 | 6 | 0 | 1 |
| 7 | -0.765 | 5 | 0 | 1 |
| 8 | -1.221 | 8 | 0 | 1 |